Method for Reducing Wafer Shape and Thickness Measurement Errors Resulted From Cavity Shape Changes

ABSTRACT

Methods and systems for reducing wafer shape and thickness measurement errors resulted from cavity shape changes are disclosed. Cavity calibration process is performed immediately before the wafer measurement. Calibrating the cavity characteristics every time the method is executed reduces wafer shape and thickness measurement errors resulted from cavity shape changes. Additionally or alternatively, a polynomial fitting process utilizing a polynomial of at least a second order is utilized for cavity tilt estimation. High order cavity shape information generated using high order polynomials takes into consideration cavity shape changes due to temperature variations, stress or the like, effectively increases accuracy of the wafer shape and thickness information computed.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit under 35 U.S.C. §119(e) ofU.S. Provisional Application Ser. No. 61/587,394, filed Jan. 17, 2012.Said U.S. Provisional Application Ser. No. 61/587,394 is herebyincorporated by reference in its entirety.

TECHNICAL FIELD

The disclosure generally relates to the field of measuring technology,particularly to methods for reducing wafer shape and thicknessmeasurement errors resulted from cavity shape changes.

BACKGROUND

Thin polished plates such as silicon wafers and the like are a veryimportant part of modern technology. A wafer, for instance, refers to athin slice of semiconductor material used in the fabrication ofintegrated circuits and other devices. Other examples of thin polishedplates may include magnetic disc substrates, gauge blocks and the like.While the technique described here refers mainly to wafers, it is to beunderstood that the technique also is applicable to other types ofpolished plates as well.

Generally, certain requirements may be established for the flatness andthickness uniformity of the wafers. There exist a variety of techniquesto address the measurement of shape and thickness variation of wafers.One such technique is disclosed in U.S. Pat. No. 6,847,458, which iscapable of measuring the surface height on both sides and thicknessvariation of a wafer. It combines two phase-shifting Fizeauinterferometers to simultaneously obtain two single-sided distance mapbetween each side of a wafer and corresponding reference flats, andcomputes thickness variation and shape of the wafer from the data andcalibrated distance map between two reference flats. However, thistechnique only reduces errors caused by the tilt or the first order ofcavity shape changes. It cannot remove errors caused by higher ordershape changes of the cavity, including changes caused by temperature orstress.

Therein lies a need for methods for reducing wafer shape and thicknessmeasurement errors resulted from cavity shape changes without theaforementioned shortcomings.

SUMMARY

The present disclosure is directed to a method for measuring thethickness variation and shape of wafers. The method includes:calibrating characteristics of a cavity formed between reference flatsin two opposing interferometer channels; placing a wafer in the cavityimmediately upon completion of calibrating the cavity characteristics;synchronizing the interferograms in the two interferometer channels bysupplying the light from a single wavelength tunable laser light source,the output beam of which is split by a beam splitter to propagate toboth interferometer channels; measuring the cavity characteristics ofthe reference flats forming the cavity providing an oversized field ofview of the wafer in the cavity for obtaining a difference map in thecavity areas outside of the wafer and measuring the cavity tilt of thereference flats; and determining the thickness variations of the waferbased on the cavity characteristic measurements and the cavity tilt ofthe reference flats. In accordance with the present disclosure, thecavity characteristics are calibrated every time the method is executedto reduce wafer shape and thickness measurement errors resulted fromcavity shape changes.

A further embodiment of the present disclosure is also directed to amethod for measuring the thickness variation and shape of wafers. Themethod includes placing a wafer in a cavity formed between referenceflats in two opposing interferometer channels; synchronizing theinterferograms in the two interferometer channels by supplying the lightfrom a single wavelength tunable laser light source, the output beam ofwhich is split by a beam splitter to propagate to both interferometerchannels; measuring the cavity characteristics of the reference flatsforming the cavity providing an oversized field of view of the wafer inthe cavity for measuring the cavity tilt of the reference flats;obtaining a difference map in the cavity areas outside of the wafer;surface fitting a polynomial of at least a second order to thedifference map; utilizing coefficients of the polynomial determined bysurface fitting to provide cavity tilt estimation; and determining thethickness variations of the wafer based on the cavity characteristicmeasurements and the cavity tilt estimation. High order cavity shapeinformation generated using high order polynomials takes intoconsideration cavity shape changes due to temperature variations, stressor the like, effectively increases accuracy of the wafer shape andthickness information computed.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not necessarily restrictive of the present disclosure. Theaccompanying drawings, which are incorporated in and constitute a partof the specification, illustrate subject matter of the disclosure.Together, the descriptions and the drawings serve to explain theprinciples of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The numerous advantages of the disclosure may be better understood bythose skilled in the art by reference to the accompanying figures inwhich:

FIG. 1 is a diagrammatic representation of an interferometer system formeasuring shape and thickness variation of a wafer;

FIG. 2 is an illustration depicting a wafer placed in the measurementcavity of the interferometer system;

FIG. 3 is an illustration depicting a wafer surface interferogramobtained utilizing the interferometer system;

FIG. 4 is a flow diagram illustrating a method for measuring the shapeand thickness variation of a wafer;

FIG. 5 is a flow diagram illustrating a method for measuring the shapeand thickness variation of a wafer utilizing one of the cavity shapechange error reduction techniques; and

FIG. 6 is a flow diagram illustrating a method for measuring the shapeand thickness variation of a wafer utilizing another cavity shape changeerror reduction techniques.

DETAILED DESCRIPTION

Reference will now be made in detail to the subject matter disclosed,which is illustrated in the accompanying drawings.

The present disclosure is directed to reducing wafer shape and thicknessmeasurement errors resulted from cavity shape changes in waferdimensional geometry tools such as the WaferSight metrology system fromKLA-Tencor. More specifically, between the completion of systemcalibration and the time that wafer measurements are taken, changes suchas temperature variations or stress may have already caused the shape ofthe measurement cavity to change. As the demand for the measurementaccuracy of the wafer thickness variation increases dramatically, therigid body assumption of the reference flats is no long valid.Therefore, cavity shape changes need to be taken into consideration inwafer dimensional geometry tools in order to reduce wafer shape andthickness measurement errors.

Referring to FIG. 1, a block diagram depicting the measurement system100 that utilizes two Fizeau interferometers similar to that disclosedin U.S. Pat. No. 6,847,458 (the disclosure of which is incorporatedherein by reference in its entirety) is shown. As depicted in FIG. 1,the measurement system 100 is configured for measuring the shape andthickness of a wafer 60. The wafer 60 may be placed in a cavity in thecenter between two Fizeau interferometers 20 and 40. The reference flats32 and 52 of the interferometers are placed close to the wafer 60.

The measurement system 100 provides two light sources for Channel A andChannel B through fiber 22 and fiber 42 from a single illuminator 8 thatgenerates a constant power output during its wavelength tuning. In oneembodiment, the light source 24,44 provides light that passes through aquarter-wave plate 28,48 aligned at 45° to the polarization direction oflight after it is reflected from the polarizing beam splitter 26,46.This beam then propagates to the lens 30,50, where it is collimated witha beam diameter larger than the wafer diameter.

The beam then goes through transmission flat 32,52, where the centralpart of the transmitted beam is reflected at the test surface 61,62 thatforms an interferogram with the light beam reflected from the referencesurface 33,53. The outer part of the transmitted beam travels on to theopposite reference flat 52,32, where it is reflected at the referencesurface 53,33 that forms an annular shape interferogram with the lightbeam reflected from the reference surface 33,53. An interferogramdetectors (e.g., an imaging device such as a camera or the like) 36,56is utilized to record the interferograms and send the interferograms toa computer 38,58 for processing to produce the desired information suchas the shape and the thickness variation of a wafer.

Theoretically, the measurement system as described above allows thethickness variation of a wafer to be measured without the effect ofcavity shape. FIG. 2 depicts how to compute the wafer thicknessvariation that is independent of the flatness of reference flat A 52 andreference flat B 32. Let φ_(A) denote the interferogram phase related tothe optical path difference between the wafer surface t_(A) and thereference A surface r_(A). Similarly, let φ_(B) denote the interferogramphase related to the optical path difference between the wafer surfacet_(B) and the reference B surface r_(B). In addition, let φ_(C) denotethe interferogram phase related to the optical path difference betweenthe reference B surface r_(B) and the reference A surface r_(A).

It is noted that the distance d between two surfaces is related to itsinteferogram phase φ. The relationship can be express as:

$\begin{matrix}{{d\left( {x,y} \right)} = {\frac{\lambda}{4\pi}\left( {{\varphi \left( {x,y} \right)} + {2\; n\; \pi}} \right)}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

Where λ is the wavelength utilized by the Fizeau interferometers and nis an unknown constant that is independent of the location (x,y) and canbe disregarded. Thus the wafer thickness variation f(x,y) can bedetermined by:

$\begin{matrix}{{f\left( {x,y} \right)} = {{{C\left( {x,y} \right)} - {d_{A}\left( {x,y} \right)} - {d_{B}\left( {x,y} \right)}} = {\frac{\lambda}{4\pi}\varphi_{f}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

where:

φ_(f)(x,y)=φ_(C)(x,y)−φ_(A)(x,y)−φ_(B)(x,y)  Equation 3:

Since the constant 2nπ term in equation 2 is eliminated, f(x,y) obtainedis not the wafer absolute thickness but rather the wafer thicknessvariation.

It is clearly depicted in FIG. 2 and Equation 3 that the phase φ_(f) canbe computed if φ_(C), φ_(A) and φ_(B) at every location (x,y) are knownduring each measurement. However, the cavity phase φ_(C) may not bereadily available at every location (x,y) since during the wafermeasurement the cavity is blocked by the opaque measuring wafer 60. FIG.3 is an illustration depicting an exemplary interferogram 300 acquiredby the interferometer 40 (or interferometer 20). As shown in FIG. 3, thewafer surface interferogram related to φ_(A) (or φ_(B)) 302 and thecavity areas outside of the wafer (in the form of a small ring 304 inthis particular example) are obtained.

Several techniques may be utilized in order to obtain the cavity phaseφ_(C) blocked by the measuring wafer 60. For instance, as depicted inmethod 400 shown in FIG. 4, measurement of the cavity without a wafer init may be taken in step 402 to obtain a cavity phase map that is largerthan the wafer in diameter. Utilizing the techniques disclosed in U.S.Pat. No. 6,847,458, step 404 may determine the tilt between the tworeference surfaces and remove the tilt from the cavity phase map.Subsequently, the cavity areas outside of the wafer, in the form of aring in the example shown in FIG. 3, can be fitted/mapped. Morespecifically, step 406 may acquire measurement data with the waferinside the cavity, or during the wafer measurement, step 408 may computethe wafer phase map φ_(A) and φ_(B), as well as φ_(C) in the cavityareas outside of the wafer, i.e., in the form of a ring in the exampleshown in FIG. 3. Step 410 may then obtain the difference map θ(x,y), orφ_(C) minus the cavity phase determined in step 404 at this same ringarea. That is, the difference map θ(x,y) represents the cavity map atthe ring area obtained during wafer measurement minus the cavity mapsaved in the calibration process for the same area. Subsequently, step412 may fit the difference map θ(x,y) by the first order polynomial:

θ(x,y)=a ₀ +a ₁ x+a ₂ y  Equation 4:

Once coefficients a₁ and a₂ are determined by the fitting process, step414 may introduce the term a₁x+a₂y to the cavity phase map (with thetilt removed) obtained in step 404. The term a₁x+a₂y is used in thismanner to estimate the tilt for a given location (x,y). Together withthe cavity phase map obtained in step 404, φ_(C) may be calculated forevery location (x,y), and the wafer thickness variation may becalculated in step 416 using Equation 3 defined above.

In practice, while steps 406 through 416 are executed for every wafermeasurement, steps 402 and 404 are considered calibration steps and areperformed periodically (e.g., every 24 hours or the like). It isconfigured in this manner in order to provide higher throughput, and isbased on the understanding that both reference flats 32 and 52 that formthe cavity are rigid bodies so that add tilt to the saved map at abovestep 406 should be sufficient. However, as the demand for themeasurement accuracy of the wafer thickness variation increasesdramatically, the rigid body assumption of the reference flats is nolong valid. More specifically, between the completion of systemcalibration (steps 402 and 404) and the time that wafer measurements aretaken (steps 406 through 416), changes such as temperature variations orstress may have already caused the shape of the measurement cavity tochange. Therefore, cavity shape changes need to be taken intoconsideration in wafer dimensional geometry tools in order to reducewafer shape and thickness measurement errors.

Two approaches may be utilized to reduce wafer shape and thicknessmeasurement errors resulted from cavity shape changes. In oneembodiment, a system calibration is performed right before the wafer isplaced into the cavity for every wafer measurement. That is, steps 402and 404 are performed immediately prior to the rest of the measurementsteps every time a new wafer is to be measured. This approach yieldsvery good measurement result because the cavity change between thecavity measurement and the wafer measurement is very small during thisshort period of time therefore and can be ignored. It is understood,however, that this approach may double the measuring time and lower thesystem throughput.

In an alternative embodiment, a higher order polynomial (greater thanthe first order) is fitted to the tilt previously described in step 412.That is, the difference map θ(x,y) is fitted by a polynomial of at leastthe second order, such as:

θ(x,y)=a ₀ +a ₁ x+a ₂ y+a ₃ x ² +a ₄ xy+a ₅ y ²+ . . .   Equation 5:

It is noted that the polynomial described above is open-ended toindicate that polynomials of higher order may be utilized withoutdeparting from the spirit and scope of the present disclosure. Usinghigher order polynomials increases accuracy but also introduces morecoefficients that need to be determined by the polynomial fittingprocess. It is contemplated that specific polynomial equations used toexpress θ(x,y) is not limited to the polynomial described above. A moregeneric high order fit can be expressed as:

θ(x,y)=Σ_(n=0) ^(∞) a _(n) g(x,y)  Equation 6:

where g(x,y) is a set of functions.

Furthermore, it is contemplated that while the description abovereferences the Cartesian coordinate system (x,y), such a coordinatesystem is merely exemplary. Other coordinate systems such as the polarcoordinate system (r,θ) may also be utilized without departing from thespirit and scope of the present disclosure.

It is also contemplated that the two approaches described above may becombined to further improve wafer measurement accuracy. Whether toimplement either one or both approaches in wafer dimensional geometrytools such as the WaferSight may be determined based upon designpreferences and requirements.

Referring now to FIG. 5, a method 500 for measuring the shape andthickness variation of a wafer utilizing the cavity shape change errorreduction technique is shown. Step 502 first calibrates the measurementsystem. In one embodiment, the phase shifting speed of theinterferograms in the two interferometer channels are first calibratedby placing a polished opaque plate in the cavity between the referenceflats 32 and 52. Alternatively, this calibration may be conducted by thecavity itself (without the polished opaque plate). Upon completion ofthe phase shift calibration, or when the phase shift between anyadjacent frames is within ±1 degree or less of its expected value suchas 90 degrees for the phase shift between any adjacent frames, thecavity characteristics of the reference flats 32 and 52 is thencalibrated with the cavity itself. Method 500 requires thecharacteristics of the reference flats 32 and 52 to be calibratedimmediately prior to the rest of the measurement steps every time a newwafer is to be measured.

Once the measurement system is calibrated, the wafer 60 that is to bemeasured is placed in the cavity in step 504. The wafer 60 may be placedin between the two Fizeau interferometers 20 and 40 (more specifically,between the reference flats 32 and 52). A holding container may beutilized to removably secure the wafer 60 when the wafer 60 is placed inthe cavity. The holding container may be configured in a manner suchthat both wafer sides 61 and 62 are minimally obscured by the holdingcontainer. While it may be beneficial to place the wafer 60 in thecenter of the cavity (i.e., the distance between the reference surface33 and 61 is substantially equal to the distance between the referencesurface 53 and 62), such a placement is not required. It is contemplatedthat if the wafer 60 is placed in an off-center position and/or rotatedfrom its expected position inside the cavity, image processingalgorithms associated with the imaging systems 36 and 56 may be utilizedto compensate for such an off-center placement and/or rotation.

Step 506 may then acquire two sets of intensity frames that recordinterferograms in Channel A and Channel B by varying the wavelength ofthe light source 8. Step 508 may extract phases of interferograms fromthese intensity frames, and step 510 may compute the shape and thicknessinformation based on the phases and phase shifts of interferogramsextracted in step 508. In one embodiment, the shape and thicknessinformation may be computed in a manner similar to that disclosed inU.S. Pat. No. 6,847,458. For instance, let A denote the phase ofinterferogram formed by reference flat 32 and wafer surface 61, let Bdenote the phase of interferogram formed by the reference flat 53 andwafer surface 62, and let C denote the phase of interferogram formed bythe cavity of two reference flats 32 and 53. Thus A provides informationregarding the height of the wafer surface 61, B provides informationregarding the height of the wafer surface 62, and C−(A+B) providesinformation regarding the thickness variation of the wafer 60.

Referring now to FIG. 6, a method 600 for measuring the shape andthickness variation of a wafer utilizing another cavity shape changeerror reduction technique is shown. Step 602 first calibrates themeasurement system similar to step 502 described above. However, method600 does not require the characteristics of the reference flats 32 and52 to be calibrated immediately prior to the rest of the measurementsteps every time a new wafer is to be measured.

Similar to method 500 described above, once the measurement system iscalibrated, the wafer 60 that is to be measured may be placed in thecavity in step 604, and step 606 may then acquire two sets of intensityframes that record interferograms in Channel A and Channel B by varyingthe wavelength of the light source 8. Step 608 may subsequently extractphases of interferograms from these intensity frames.

Different from method 500 is the generation of high order cavity shapeinformation in step 610. High order cavity shape information generatedusing high order polynomials as described above takes into considerationcavity shape changes due to temperature variations, stress or the like,effectively increases accuracy of the wafer shape and thicknessinformation computed in step 612.

It is contemplated that while the examples above referred to wafermetrology measurements, the systems and methods in accordance with thepresent disclosure are applicable to other types of polished plates aswell without departing from the spirit and scope of the presentdisclosure. The term wafer used in the present disclosure may include athin slice of semiconductor material used in the fabrication ofintegrated circuits and other devices, as well as other thin polishedplates such as magnetic disc substrates, gauge blocks and the like.

It is to be understood that the present disclosure may be implemented informs of a software/firmware package. Such a package may be a computerprogram product which employs a computer-readable storage medium/deviceincluding stored computer code which is used to program a computer toperform the disclosed function and process of the present disclosure.The computer-readable medium may include, but is not limited to, anytype of conventional floppy disk, optical disk, CD-ROM, magnetic disk,hard disk drive, magneto-optical disk, ROM, RAM, EPROM, EEPROM, magneticor optical card, or any other suitable media for storing electronicinstructions.

The methods disclosed may be implemented as sets of instructions,through a single production device, and/or through multiple productiondevices. Further, it is understood that the specific order or hierarchyof steps in the methods disclosed are examples of exemplary approaches.Based upon design preferences, it is understood that the specific orderor hierarchy of steps in the method can be rearranged while remainingwithin the scope and spirit of the disclosure. The accompanying methodclaims present elements of the various steps in a sample order, and arenot necessarily meant to be limited to the specific order or hierarchypresented.

It is believed that the system and method of the present disclosure andmany of its attendant advantages will be understood by the foregoingdescription, and it will be apparent that various changes may be made inthe form, construction and arrangement of the components withoutdeparting from the disclosed subject matter or without sacrificing allof its material advantages. The form described is merely explanatory.

What is claimed is:
 1. A method for measuring the thickness variationand shape of wafers, the method comprising: calibrating characteristicsof a cavity formed between reference flats in two opposinginterferometer channels; placing a wafer in the cavity immediately uponcompletion of calibrating the cavity characteristics; synchronizing theinterferograms in the two interferometer channels by supplying the lightfrom a single wavelength tunable laser light source, the output beam ofwhich is split by a beam splitter to propagate to both interferometerchannels; measuring the cavity characteristics of the reference flatsforming the cavity providing an oversized field of view of the wafer inthe cavity for obtaining a difference map in the cavity areas outside ofthe wafer and measuring the cavity tilt of the reference flats; anddetermining the thickness variations of the wafer based on the cavitycharacteristic measurements and the cavity tilt of the reference flats,wherein the cavity characteristics are calibrated every time the methodis executed to reduce wafer shape and thickness measurement errorsresulted from cavity shape changes.
 2. The method of claim 1, whereinthe step of measuring the cavity tilt includes a step of measuring thecavity tilt substantially simultaneously with mapping of oppositesurfaces of the wafer.
 3. The method of claim 1, wherein obtaining adifference map in the cavity areas outside of the wafer furtherincludes: obtaining a cavity map during wafer measurement for the cavityareas outside of the wafer based on the oversized field of view of thewafer in the cavity; and calculating the difference map by subtracting amap of the cavity areas outside of the wafer obtained during cavitycalibration from the cavity map obtained during wafer measurement. 4.The method of claim 1, wherein measuring the cavity characteristics ofthe reference flats further includes: surface fitting a polynomial tothe difference map.
 5. The method of claim 4, wherein the polynomial isa first order polynomial.
 6. The method of claim 4, wherein thepolynomial is of at least a second order.
 7. A method for measuring thethickness variation and shape of wafers, the method comprising: placinga wafer in a cavity formed between reference flats in two opposinginterferometer channels; synchronizing the interferograms in the twointerferometer channels by supplying the light from a single wavelengthtunable laser light source, the output beam of which is split by a beamsplitter to propagate to both interferometer channels; measuring thecavity characteristics of the reference flats forming the cavityproviding an oversized field of view of the wafer in the cavity formeasuring the cavity tilt of the reference flats; obtaining a differencemap in the cavity areas outside of the wafer; surface fitting apolynomial of at least a second order to the difference map; utilizingcoefficients of the polynomial determined by surface fitting to providecavity tilt estimation; and determining the thickness variations of thewafer based on the cavity characteristic measurements and the cavitytilt estimation.
 8. The method of claim 7, wherein the step of measuringthe cavity tilt includes a step of measuring the cavity tiltsubstantially simultaneously with mapping of opposite surfaces of thewafer.
 9. The method of claim 7, further including the step ofcalibrating characteristics of the cavity prior to placing the wafer inthe cavity.
 10. The method of claim 9, wherein obtaining a differencemap in the cavity areas outside of the wafer further includes: obtaininga cavity map during wafer measurement for the cavity areas outside ofthe wafer based on the oversized field of view of the wafer in thecavity; and calculating the difference map by subtracting a map of thecavity areas outside of the wafer obtained during cavity calibrationfrom the cavity map obtained during wafer measurement.
 11. The method ofclaim 9, wherein the cavity characteristics are calibrated every timethe method is executed to reduce wafer shape and thickness measurementerrors resulted from cavity shape changes.
 12. An apparatus formeasuring the thickness variation and shape of a polished opaque plate,the apparatus comprising: first and second spaced apart reference flatshaving corresponding first and second parallel reference surfacesforming a cavity therebetween for placement of the polished opaqueplate, the reference surfaces providing an oversized field of view ofthe plate in the cavity; first and second interferometer devices locatedon diametrically opposite sides of the cavity to map the opposite firstand second surfaces of the polished opaque plate; a light sourceoptically coupled to the first and second interferometer devices, thelight source comprising an illuminator configured for producing light ofmultiple wavelengths and an optical amplitude modulator configured forstabilizing power of the light produced by the illuminator; first andsecond interferogram detectors; and at least one processing unit coupledto receive the outputs of the first and second interferogram detectorsfor determining thickness variations of the plate, the at least oneprocessing unit configured for: measuring the cavity characteristics ofthe reference flats; obtaining a difference map in the cavity areasoutside of the plate; surface fitting a polynomial to the differencemap; utilizing coefficients of the polynomial determined by surfacefitting to provide cavity tilt estimation; and determining the thicknessvariations of the plate based on the cavity characteristic measurementsand the cavity tilt estimation.
 13. The apparatus of claim 12, whereinthe polynomial is a first order polynomial.
 14. The apparatus of claim12, wherein the polynomial is of at least a second order.
 15. Theapparatus of claim 12, wherein measuring the cavity characteristicsincludes a step of measuring the cavity tilt substantiallysimultaneously with mapping of opposite surfaces of the plate.
 16. Theapparatus of claim 12, wherein the at least one processing unit isfurther configured for calibrating characteristics of the cavity priorto placing the plate in the cavity.
 17. The apparatus of claim 14,wherein obtaining a difference map in the cavity areas outside of theplate further includes: obtaining a cavity map during plate measurementfor the cavity areas outside of the plate based on the oversized fieldof view of the plate in the cavity; and calculating the difference mapby subtracting a map of the cavity areas outside of the plate obtainedduring cavity calibration from the cavity map obtained during platemeasurement.
 18. The apparatus of claim 14, wherein the cavitycharacteristics are calibrated every time measurement is taken to reduceplate shape and thickness measurement errors resulted from cavity shapechanges.
 19. The apparatus of claim 14, wherein the first and secondinterferometer devices are Fizeau interferometers.
 20. The apparatus ofclaim 14, wherein the apparatus is configured for measuring thethickness variation and shape of a silicon wafer.